Question

Smith and Jones each write the same multiple choice test. The test has 5 questions, and each question has 5 answers (exactly one of which is right). Smith and Jones are not very well prepared for the test and they answer the questions randomly.

(i) FInd the probability that they both get the same number of answers correct. (ANSWER: 0.3198)

(ii) Find the probability that their papers are identical, assuming that they have answered independently of one another. (ANSWER: 0.00032)

Answer 1

There are 5 possible answers for each question. The probability that a randomly selected answer for a question is correct is p=1/5=0.20

There are 5 questions. Let X be the number of questions answered correctly by Smith. X has a Binomial distribution with parameters, number of trials n=5 and success probability (probability of answering the question correctly) p=0.20

$\begin{align*} P(X=x)&=\binom{n}{x}p^x(1-p)^{n-x}\\ &=\binom{5}{x}0.20^x(1-0.20)^{5-x}\\ &=\frac{5!}{x!(5-x)!}0.20^x0.80^{5-x}\\ \end{align*}$

Let Y be the number of questions answered correctly by Jones. Y has a Binomial distribution with parameters, number of trials n=5 and success probability (probability of answering the question correctly) p=0.20

$\begin{align*} P(Y=y)&=\binom{n}{y}p^y(1-p)^{n-y}\\ &=\binom{5}{y}0.20^y(1-0.20)^{5-y}\\ &=\frac{5!}{y!(5-y)!}0.20^y0.8^{5-y}\\ \end{align*}$

i) Since Smith and Jones answer independently of each other, The joint probability of X,Y is given by

$\begin{align*} P(X=x,Y=y)&=P(X=x)P(Y=y) \end{align*}$

The probability that both get the same number of answers correct is the probability that X=Y

$\begin{align*} P(X=Y)&=\sum_{x=0}^5P(X=x,Y=x)\\ &=\sum_{x=0}^5P(X=x)P(Y=x)\\ &=P(X=0)P(Y=0)+P(X=1)P(Y=1)+P(X=2)P(Y=2)+P(X=3)P(Y=3)+P(X=4)P(Y=4)+P(X=5)P(Y=5)\\ &=\frac{5!}{0!(5-0)!}0.2^00.8^{5-0}\times \frac{5!}{0!(5-0)!}0.2^00.8^{5-0}+...+\frac{5!}{5!(5-5)!}0.2^50.8^{5-5}\times \frac{5!}{5!(5-5)!}0.2^50.8^{5-5}\\ &=0.32768\times0.32768+ 0.4096\times0.4096+ 0.2048\times0.2048+ 0.0512\times0.0512+ 0.0064\times0.0064+ 0.00032\times0.00032\\ &=0.3198 \end{align*}$ans: The probability that both get the same number of answers correct is 0.3198

ii) Each question can be answered in 5 different ways (there are 5 possible answers).

The total number of ways to answer 5 questions is $\begin{align*} 5^5=3125 \end{align*}$

the 2 papers are identical if the answer to each one of 5 questions match. There is only 1 possible way for this to happen.

The probability that their papers are identical is

$\begin{align*} Pr(\text{Identical})&=\frac{\text{Number of ways in which the 2 papers can be identical}}{\text{Total number of ways to answer 5 questions}}\\ &=\frac{1}{3125}\\ &=0.00032 \end{align*}$

ans: The probability that their papers are identical is 0.00032